In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 10081015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.

No abstract uploaded!

An almost K\"ahler structure on a symplectic manifold (N,ω) consists of a Riemannian metric g and an almost complex structure J such that the symplectic form ω satisfies ω(⋅,⋅)=g(J(⋅),⋅) . Any symplectic manifold admits an almost K\"ahler structure and we refer to (N,ω,g,J) as an almost K\"ahler manifold. In this article, we propose a natural evolution equation to investigate the deformation of Lagrangian submanifolds in almost K\"ahler manifolds. A metric and complex connection $\hn$ on TN defines a generalized mean curvature vector field along any Lagrangian submanifold M of N . We study the evolution of M along this vector field, which turns out to be a Lagrangian deformation, as long as the connection $\hn$ satisfies an Einstein condition. This can be viewed as a generalization of the classical Lagrangian mean curvature flow in K\"ahler-Einstein manifolds where the connection $\hn$ is the Levi-Civita connection of g . Our result applies to the important case of Lagrangian submanifolds in a cotangent bundle equipped with the canonical almost K\"ahler structure and to other generalization of Lagrangian mean curvature flows, such as the flow considered by Behrndt \cite{b} in K\"ahler manifolds that are almost Einstein.

Let M be a closed surface diffeomorphic to S2 endowed with a Riemannian metric. Denote the diameter of M by d. We prove that for every x 2 M and every positive integer k there exist k distinct geodesic loops based at x of length  20kd.

We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach space$X$is reflexive if and only if it admits and equivalent norm such that there exists$x$_{0}≠0 in$X$and a weak-^{*}-open subset$A$of the dual space, satisfying that$x$^{*}⊗$x$_{0}attains its numerical radius. for each$x$^{*}in$A$.